[en] The open p-adic string world sheet is a coset space F=T/Γ, where T is the Bruhat-Tits tree for the p-adic linear group GL(2,Q_p) and Γ is contained in PGL(2,Q_p) is some Schottky group. The boundary of this world sheet corresponds to a p-adic Mumford curve of finite genus. The string dynamics is governed by the local action on F, with the fields taking values in a compact group G. We find the correlation functions and partition functions for the curves of arbitrary genus and G=U(1)^xD (D-dimensional torus). (author). 21 refs